Which is a equation defining just the standard three dimensional acceleration.

And when the particle becomes very, acquires a very high velocity due

to this acceleration, it's where a line Is changed according to this equation.

In fact this equation can be obtained from this equation by easy means.

You see that zed 1 from here is just

a square root of 1 over a squared

plus zed 0 squared minus 1 over a.

And if this guy is much smaller than this guy,

we can taylor expand the square root, and that's how we get this expression.

This is another way to obtain the same equation.

So obviously we obtain accelerated motion.

Now we want to transfer to the reference system of this kind of motion.

To the reference system of a homogeneously constantly accelerating observers.

How do we do that?

Well that's what we're going to do right now, but

let me just stressed that motion along this hyperbola from the point of

view of the one who's moving along this hyperbola is absolutely homogeneous.

The one who moves along this hyperbola cannot distinguish

its point from any other point.

It's indistinguishable.

It's only from this picture this point is different from the other etcetera.

But if you are sitting in the reference system, which moves with

constant acceleration, this point is no different from this one or from this one.

In fact, because like if you change your reference system, which is inertial and

moving with instantly moving with this particle at this point.

Well the hyperbola just turns around, it transforms and this point is transferred

to this one and corresponds to the particle, which is having zero velocity.

But again the motion along this hyperbola is homogeneous.

And it's invariant on the proper time translations, and time reversal.

And as a result, we expect that the metric

in the co-moving non-inertial reference frame will be stationary.

So let us see that explicitly.

Inspired by these formulas, by these

equations, now we will write our information.

Let us write the full and current information.

Namely, let us take x 0 to be low,